In this article, we consider the wave equation on a domain of Rn with Lipschitz
boundary. For every observable subset Γ of the boundary ∂Ω (endowed with the usual Hausdorff measure
Hn −
1 on ∂Ω), the observability constant provides an account
for the quality of the reconstruction in some inverse problem. Our objective is here to
determine what is, in some appropriate sense, the best observation domain. After having
defined a randomized observability constant, more relevant tan the usual
one in applications, we determine the optimal value of this constant over all possible
subsets Γ of prescribed area
Hn −
1(Γ) = LHn −
1(∂Ω), with L ∈ (0,1), under
appropriate spectral assumptions on Ω. We compute the maximizers of a relaxed version of the problem, and
then study the existence of an optimal set of particular domains Ω. We then define and study an approximation
of the problem with a finite number of modes, showing existence and uniqueness of an
optimal set, and provide some numerical simulations.

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